Dyson Brownian motion (DBM) is a powerful tool for studying the spectral universality of random matrices: it interpolates general Wigner matrices with Gaussian ensembles through a stochastic flow. Homogenization theory analyzes the leading-order behaviour of the difference between two coupled DBMs. Recently, Bourgade (2021) developed a novel approach to homogenization, yielding optimal estimates in the bulk of the spectrum. We extend the result throughout the spectrum, including up to the spectral edges. As an application, we show that the Kolmogorov-Smirnov distance of the distribution of the gap between the largest two eigenvalues of a generalized Wigner matrix (with smooth entry distribution) and its GOE/GUE counterpart is O(N^{-1+\epsilon}). We also construct Wigner matrices so that the analogous Kolmogorov-Smirnov distance for the distribution of the largest eigenvalue is bounded below by O(N^{-1/3+\epsilon}), demonstrating the quantitative edge universality result of Schnelli and Xu (2022) is optimal. Joint work with Benjamin Landon.